Kristen Michaelis explains this intolerance very clearly on the website Food Renegade:

First, let’s be clear about what gluten intolerance is.
It isn’t a food allergy. It’s a physical condition in your gut.
Basically, undigested gluten proteins (prevalent in wheat and other
grains) hang out in your intestines and are treated by your body like a
foreign invader, irritating your gut and flattening the microvilli along
the small intestine wall. Without those microvilli, you have
considerably less surface area with which to absorb the nutrients from
your food. This leads sufferers to experience symptoms of malabsorption,
including chronic fatigue, neurological disorders, nutrient
deficiencies, anemia, nausea, skin rashes, depression, and more.

If you remove gluten from the diet, the gut heals and the myriad of symptoms disappears. (source)

You can’t throw a shoe in the air at Whole Foods or Trader Joes
without it landing on someone looking at “gluten free” products. But the
question many nutrition professionals are asking is this:

Is gluten intolerance just the next fad diet or is wheat really toxic?